My question is the following: if we had the trajectory of a particle eventually reaching a point of a rotation axis $ \vec{u} $ (take that as being the z-axis for convenience) by an angle $ s $, would Noethers Theorem still give a conserved quantity?

More specifically (let me go through the calculations and details first)

Working out the conserved quantity, we get that the z-component of angular momentum $ L_z = m \dot{y}(t) x(t) - m \dot{x}(t) y(t) $ is conserved for any path $ (x(t),y(t),z(t)) $.

The problem:
If this trajectory would include any point on the rotation axis z, $ h^s(q_i) $ would be 0 there and so by conservation, valued 0 all along the path.
However, we know that angular momentum is conserved.
So, in all rigor - is this inconsistency amendable or a sign of some bigger problem?

2 Answers
2

There is no inconsistency. $L_z$ is conserved, but $h^s$ is not; hence it is not a paradox that $h^s$ has no action on the trajectory at a certain time and nontrivial action at others.

On the other hand, if the trajectory ever does cross the $z$ axis, then $L_z=m(\dot y x-\dot x y)$ will also vanish. Since $L_z$ is conserved, you can conclude that $L_z\equiv0$ for all time. This means that the motion of the particle will be constrained to the plane spanned by the $z$ axis and the particle's velocity when it crosses the $z$ axis.

To prove this, you can see the equation $L_z=0$ as a differential equation, which you can phrase as
$$\frac{\dot y} y=\frac{\dot x}x,$$
and which integrates easily to
$$\log(y)=\int\frac{\text dy}{y}=\int\frac{\text dx}{x}=\log(x)+C$$
so $x$ and $y$ are proportional, and therefore constrained to a plane. This is what you can conclude from the particle ever crossing the $z$ axis in a system with rotational symmetry about that axis.

Working out the conserved quantity, we get that the $z$-component of angular momentum $ L_z = m x(t)\dot{y}(t) - m y(t)\dot{x}(t)$ is conserved for any path $(x(t),y(t),z(t))$.

Well, it should be stressed that the conservation law in Noether's theorem is an so-called on-shell conservation law. It does not hold for any curved (off-shell) path that one could cook up. It only holds for classical paths, i.e. solutions to Euler-Lagrange equations. For OP's example of a free particle this means constant velocity along a straight line. If such a classical trajectory intersects the $z$-axis (at one point), then the particle would indeed have angular momentum $L_z(t)=0$ for all $t$.